On the dependence of the Berry–Esseen bound on dimension

Abstract

Let X be a random vector with values in R d . Assume that X has mean zero and identity covariance. Write β= E |X| 3 . Let S n be a normalized sum of n independent copies of X. For Δ n= sup A∈ C | P{S n∈A}−ν(A)| , where C is the class of convex subsets of R d , and ν is the standard d-dimensional normal distribution, we prove a Berry–Esseen bound Δ n⩽400d 1/4β/ n . Whether one can remove or replace the factor d 1/4 by a better one (eventually by 1), remains an open question.